Integrating xcosx dx: Why No C?

In summary, when using integration by parts to find the integral of xcosx dx, we assume that cosx dx = du and integrate both sides to find u. We can take the integration constant into account, but it is usually better to assume it as zero in the intermediate steps. This is because it will cancel out in the final result. Another approach is to let u = x and dv = Cos x, or for an integrand with x raised to a power, u = x^n and dv = Cos x. Tabular Integration by Parts is also a helpful method for this type of integral.
  • #1
MHD93
93
0
Hi

When we use the integration by parts to find the integral of xcosx dx, we assume that cosx dx = du and integrate both sides to find u
When we integrate
du = cosxdx
we find that u = sinx
the question is why don't we take the integration constant C into account?
 
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  • #2
Mohammad_93 said:
the question is why don't we take the integration constant C into account?

You can take it into account if you want, you'll get the same result (except for a final integration constant of course), so it's usually better to assume this constant zero in the intermediate steps.
 
  • #3
so it's usually better to assume this constant zero in the intermediate steps.

But it may or mayn't be zero, therefore the final integration is different if it's not zero.
 
  • #4
I indeed am in need of your help
 
  • #5
The point is that it cancels out. Without the integration constant:
[tex]\int{udv} = uv-\int{vdu}[/tex]
With the integration constant:
[tex]\intudv = u(v+C)-\int{(v+C)du} = uv+uC-\int{vdu}-C\int{du} = uv +uC-\int{vdu}-uC = uv-\int{vdu}[/tex]
 
  • #6
Wow, that's real helpful, thank you
 
  • #7
Mohammad_93 said:
Hi

When we use the integration by parts to find the integral of xcosx dx, we assume that cosx dx = du and integrate both sides to find u When we integrate du = cosxdx we find that u = sinx

For this integrand it is much better to let u = x and dv = Cos x
Your choice will work too, but if x is raised to a power the clear choice is
u = x^n and dv = Cos x
This integral is also best done with Tabular Integration By Parts
 

1. Why is "C" not included in the integration of xcosx dx?

When integrating a function, the constant of integration (C) is typically added at the end to account for all possible solutions. However, in the case of xcosx dx, the constant is not included because the integral is definite, meaning it has specific limits of integration. Therefore, the constant is not necessary as it would be eliminated when evaluating the definite integral.

2. Can "C" be added to the integral of xcosx dx?

Technically, yes, you can add the constant of integration to the integral of xcosx dx. However, since the integral is definite, the addition of the constant would not change the result of the integral. It would simply be cancelled out when evaluating the integral with the given limits of integration.

3. How do I solve the integral of xcosx dx?

The integral of xcosx dx can be solved using integration by parts, where u = x and dv = cosx dx. This method involves using the product rule of differentiation and then integrating the resulting equation. Another approach is to use trigonometric identities to rewrite cosx in terms of sine and cosine, and then use substitution to solve the resulting integral.

4. Is there a shortcut to solving the integral of xcosx dx?

Unfortunately, there is no shortcut or direct formula for solving the integral of xcosx dx. It requires the use of integration techniques such as integration by parts or trigonometric substitution. However, with practice and familiarity with these techniques, the process can become quicker and easier.

5. What are the applications of integrating xcosx dx?

The integral of xcosx dx has various applications in physics, engineering, and mathematics. It can be used to determine the area under a curve, calculate the work done by a variable force, and solve differential equations. It also has applications in real-life scenarios such as predicting the motion of a pendulum or analyzing the vibrations of a guitar string.

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